Algebraic rate of decay for the excess free energy and stability of fronts for a nonlocal phase kinetics equation with a conservation law. I

This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v(0), of a front that is sufficiently small in L-2 norm, and sufficiently localized that integral x(2)v(0)(x(2)) dx < infinity, yields a solution that relaxes to another front, selected by a conservation law, in the L-1 norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L-2 norm and the rate of decrease of the excess free energy, Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.